
%!TEX program = xelatex
%!TEX TS-program = xelatex
%!TEX encoding = UTF-8 Unicode

\documentclass[10pt]{article} 

\input{wang_preamble.tex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{titling}
\setlength{\droptitle}{-2cm}   % This is your set screw

%%文档的题目、作者与日期
\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{实变函数练习1.1-1.2}
%\date{\vspace{-3ex}}
\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
%\date{2023 年 10 月 31 日}
%\date{March 9, 2021}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 01
用集合的语言描述函数的性质：函数 $f(x)$ 在实数轴上有定义，且在 $[a,b]$ 上有上界 $M$.

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 02
用集合的语言描述函数的性质：函数 $f(x)$ 在实数轴上有定义，且在点 $x_0\in\mathbb{R}$ 连续。

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 03
证明：对任意两个实数 $a<b$, 成立 $$(a,b) = \bigcup\limits_{n=1}^\infty \left[ a+\frac{1}{n}, b-\frac{1}{n} \right]. $$ 

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 04
使用集合的运算语言，叙述有限覆盖定理和区间套定理。

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 05
证明德摩根公式：设 $\{A_\alpha:\alpha\in\Lambda\}$ 是全集 $S$ 的一族子集，则有
\begin{eqnarray*}
S - \bigcup\limits_{\alpha\in \Lambda} A_\alpha &=& \bigcap\limits_{\alpha\in \Lambda} (S - A_\alpha), \\ 
S - \bigcap\limits_{\alpha\in \Lambda} A_\alpha &=& \bigcup\limits_{\alpha\in \Lambda} (S - A_\alpha).  
\end{eqnarray*}


\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 06
设 $\{f_n(x):n\in\mathbb{N}\}$ 是定义在 $E$ 上的函数列。则有
$$\{x\in E: \lim\limits_{n\to\infty} f_n(x) = 0\} = \bigcap\limits_{\varepsilon\in\mathbb{R}^+} 
\bigcup\limits_{N=1}^{\infty} \bigcap\limits_{n=N}^{\infty} \{x\in E: |f_n(x)|<\varepsilon\}. $$

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 07
设 $A_n$ 是下述点集，
\begin{eqnarray*}
A_{2m+1} &=& \left[ 0,2-\frac{1}{2m+1} \right], m=0,1,2,\cdots, \\ 
A_{2m} &=& \left[ 0,1+\frac{1}{2m} \right], m=1,2,3,\cdots.
\end{eqnarray*}
计算集合序列 $A_1,A_2,\cdots,A_n,\cdots$ 的上极限和下极限。

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 08
证明可以使用交集和并集来表示集合序列的上下极限：
\begin{eqnarray*}
\varlimsup\limits_{n\to\infty} A_n &=&  \bigcap\limits_{n=1}^{\infty} \bigcup\limits_{m=n}^{\infty} A_m, \\ 
\varliminf\limits_{n\to\infty} A_n &=&  \bigcup\limits_{n=1}^{\infty} \bigcap\limits_{m=n}^{\infty} A_m. 
\end{eqnarray*}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 09
证明：若 $\lim\limits_{n\to\infty} a_n = a$ 则  
$$\{a\} = \bigcap\limits_{\varepsilon>0}\varliminf\limits_{n\to\infty} \{x: |x-a_n|<\varepsilon\} .$$

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %Problem 10
设 $f(x)$ 是定义在 $E$ 上的有限函数，记 $F_n = \{x\in E: |f(x)|\ge \frac{1}{n}\}$.
证明 $\{F_n\}$ 是增加集列，且有 
$$\lim\limits_{n\to\infty} F_n = \bigcup\limits_{n=1}^{\infty} F_n = \{x\in E: f(x)\neq 0\}. $$

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{enumerate}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


